Q:
How long would it take for scientists to notice the effects of the NGC 6240 merger in our galaxy and/or in other galaxies?

A:
In the case of NGC 6240, the predicted merger of supermassive black holes has likely already occurred, perhaps 250 million years ago in NGC 6240 time. But we won't know about it for another 150 million years! See the discussion and time line at http://chandra.harvard.edu/photo/cosmic_lookback.html

The changes in the distance between points due to the gravitational waves from a distant black hole merger will be so small that it will take an extremely sensitive detector to measure them.